theory of equations, the calculus of variations, the theory of probability, the geometry of surfaces, and the subject of infinite series. Like Sir Isaac Newton, he at times displayed a disinclination to enter upon a prompt publication of his scientific deductions. As a consequence of this, others rediscovered and published results which Gauss might have claimed for himself. Thus it is now known that some of the discoveries on elliptic functions made by Abel and Jacobi had been worked out by Gauss thirty years earlier but not published. According to Professor Felix Klein, some Gaussian manuscripts reveal a knowledge of the fundamental ideas of quaternions, a subject fully elaborated later by the genius of the Irish astronomer, Sir William Rowan Hamilton. Perhaps the most striking case of loss of priority of discovery due to failure to place his results at the disposal of the general scientific public, is that of non-euclidean geometry. For many years Gauss permitted his mind to dwell upon the subtle subject of parallel lines, and he reached some exceedingly original results. But he did not write down in full what he had worked out in his mind, and nothing was published by him on this topic. Off and on he would touch upon this subject in letters to scientific friends. He expressed to them his intention not to allow any part of this research to reach the general public during his lifetime. On January 27, 1829, he wrote to Bessel: "Probably I shall not be ready for a long time yet, to prepare for publication my very extensive researches on this subject and perhaps this will not happen during my lifetime, for I would dread the clamor of the Boeotians, were I to speak out in full." Imagine his surprise when the Hungarian Wolfgang Bolyai, a close friend of his during their student days at the university, sent a printed document of twenty-six pages written by Wolfgang's son, John Bolyai, in which the young Bolyai had worked out with wonderful clearness and originality the fundamental propositions of non-euclidean geometry. Gauss saw at once that he had been anticipated. How did the world-renowned mathematician of Gottingen behave toward the young and unknown Hungarian? Students of scientific history know that on questions of priority of discovery many a bitter battle has been fought. Scientific men are only human, and they frequently fail to see the full merits of rival claimants. But Gauss showed himself as generous as a man as he was great as a scientist. After reading John Bolyai's published dissertation, he wrote to his friend Gerling as follows (February 1-4, 1832): "I consider this young geometer v. Bolyai a genius of the first rank." To his old friend Wolfgang Bolyai, Gauss wrote (March 6, 1832) in this manner:
- Professor P. G. Tait declared that Klein is mistaken and that the Gaussian restricted forms of linear and vector operators do not constitute an invention of quaternions. Klein's article is in Math. Annalen, LI., 1898. A note by Tait appeared in Proc. of the Royal Soc. of Edinburgh, December 18, 1899.